Journal of Applied Mathematics

Volume 2014 (2014), Article ID 187037, 11 pages

http://dx.doi.org/10.1155/2014/187037

Research Article

Exponential Stability of Stochastic Differential Equation with Mixed Delay

Wenli Zhu, Jiexiang Huang, Xinfeng Ruan, and Zhao Zhao

School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received 24 November 2013; Revised 21 January 2014; Accepted 21 January 2014; Published 4 March 2014

Academic Editor: Oluwole Daniel Makinde

Copyright © 2014 Wenli Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper focuses on a class of stochastic differential equations with mixed delay based on Lyapunov stability theory, Itô formula, stochastic analysis, and inequality technique. A sufficient condition for existence and uniqueness of the adapted solution to such systems is established by employing fixed point theorem. Some sufficient conditions of exponential stability and corollaries for such systems are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. In particular, our theoretical results show that if stochastic differential equation is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic differential equation with mixed delay will remain exponentially stable. Moreover, time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice.

1. Introduction

The nondeterministic (i.e., stochastic) phenomena are frequently encountered in many practical systems. These systems should be described by stochastic differential equations (SDEs for short) instead of ordinary ones. On the other hand, time delays are included in many practical systems, such as networks control systems, traffic systems, production process control systems, and population and economic dynamic systems, that is, the current and future states of the systems dependent on their departed states. In current years, the study of analysis and synthesis of stochastic time delay systems, which are described by stochastic delayed differential equations (SDDEs for short), is a popular topic in the field of control theory [1–7]. Because the existence of time delay is often the reason of instability and deteriorates the control performance, the studies on time delay systems stability and control have important theoretical and practical values.

A real dynamic system is influenced by both stochastic disturbances and time delays, so when we consider the behavior of a dynamic system, we use the stochastic delayed differential equation and the stochastic functional differential equation (SFDE for short) as modeling tools to investigate stability of stochastic dynamic systems with discrete delays or distributed delays. So far, these topics have received a lot of attention and there are so many references about them. For example, Cong [1] and Li et al. [2] obtained exponential stability conditions of linear stochastic neutral delay systems. Mao [8], Mao and Shah [9], Zhu and Hu [10], Zhu and Hu [11], S. Xie and L. Xie [12], and Zhu et al. [13] established some stability criteria of the stochastic system with discrete delays. An improved delay-dependent stability criterion is derived for stochastic delay systems by a strict LMI in [14]. Hu and Wu [15], Wu et al. [16], Yin et al. [17], and Zhou et al. [18] established some stability criteria of the stochastic system with distributed delays. However, discrete delays and distributed delays always coexist in real dynamic systems; thus, it is reasonable to consider them together and it leads us to investigate stochastic differential equations with mixed delays (SMDDEs for short).

Although stochastic differential systems with mixed delays received increasing attention recently, there is a little previous literature, as systematic research on such system has not been developed yet. For example, Zhu and Song [19] obtained some exponential stability results for a class of impulsive nonlinear stochastic differential equations with mixed delays by Razumikhin technique, but these sufficient conditions only ensure the exponential stability of the trivial solution in the mean square and did not give a bound for the time delay . Deng et al. [20] and L. Xu and D. Xu [21] focused on the corresponding study of exponential stability of neural network model. Thus, this paper aims to fill the gap in a sense. In this paper, we investigate not only the exponential stability in the mean square but also the almost surely exponential stability for a class of SMDDEs based on Lyapunov stability theory, Itô formula, stochastic analysis, inequality technique, and so on. We first consider the existence and uniqueness of the adapted solution by employing fixed point theorem. Next, some sufficient conditions of exponential stability and corollaries for stochastic differential systems with mixed delays are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of SMDDE implies the almost surely exponentially stable. The obtained results generalize and improve some recent results (for instance, [19–21]). In particular, our theoretical results show that if SDE is exponentially stable and the time delay is sufficiently small, then the corresponding SMDDE will remain exponentially stable. Moreover, the time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice. It should be mentioned that the approach provided here is different from those used in [19–21]. Finally, we present a simple example to illustrate the effectiveness of our stable results.

The rest of this paper is organized as follows. In Section 2, we give the preliminary results about SMDDEs. Main results and proofs for SMDDEs are provided in Section 3. Section 4 presents a simple example to illustrate our stable results. Section 5 lists some concluding remarks.

2. Preliminaries

Throughout this paper and unless specified, we let be an m-dimensional Brownian motion defined on a complete probability space with a natural filtration (i.e., and augmented by all the P-null sets in ). Denote by the Euclidean norm. If is a vector or matrix, its transpose is denoted by . If is a matrix, denote by the operator norm of ; that is, . is the initial path of , where is a given finite time delay and is the set of continuous functions from into . Moreover, denote by the family of -valued adapted stochastic processes , such that is -measurable and ().

We also use the notation which is -valued adapted stochastic processes s.t. .

We consider the following stochastic differential equations with mixed delays:

where and represent the nonlinear uncertainties and

represent given functional of the path segment of ; is the given averaging parameter. Furthermore, we always assume that for the stability purpose of this paper.

For simplicity, in what follows, we write sometimes, where .

To develop our theories and results, we need to introduce the following concepts and important inequalities. For stochastic system, exponential stability in mean square and almost surely exponential stability are generally used [13].

Definition 1. The trivial solution of (1) is said to be th moment exponentially stable, if there exists a positive constant such that

for any .

Especially, , and it is called mean square exponentially stable.

Definition 2. The trivial solution of (1) is said to be almost surely exponentially stable. If there exists a positive constant such that

for any .

Lemma 3 (see [22]). For any real matrices and a constant , the following matrix inequality holds:

Lemma 4 (Cauchy-Schwarz inequality). Let and be real functions which are continuous on the closed interval . Then,

3. Main Results

3.1. Existence and Uniqueness Result of the Solution for SMDDEs

We make the following assumptions for the coefficients of (1).

(H3.1)Let be fixed time duration, , , and there exists a constant such that

(H3.2).

Theorem 5. Let (H3.1) and (H3.2) hold. Then for any , (1) has a unique t-continuous adapted solution, denoted by and . So (1) has a trivial solution .

Inspired by the literature [23], we present the proof of the Theorem 5 as follows.

Proof. Let us define a norm in Banach space as follows:

Clearly it is equivalent to the original norm of . We consider

where , . Define a mapping such that . We desire to prove that is a contraction mapping under the norm . For arbitrary , set and , . Then, satisfies

Applying Itô’s formula to , we have

Integrating from to and taking the expectation in the above, we get

Lemma 3 yields

Then by (H3.1), we obtain

Thus,

where

Lemma 4 yields

It then follows from (17) that

Then,

Let ; then the above yields

That is,

This implies that is a strict contraction mapping. Then it follows from the fixed point theorem that (1) has a unique solution in . Since and satisfy (H3.1) and (H3.2), we can easily derive that and is continuous with respect to . Furthermore, by , (1) has a trivial solution .

For simplicity, in what follows we write .

3.2. Exponential Stability for SMDDEs

We make the following assumptions for the coefficients of (1).

(H3.3) There exist nonnegative constants , , for any such that

and for any such that

(H3.4) By (H3.1), one has .

In the study of mean square exponential stability, it is often to use a quadratic function as the Lyapunov function; that is, , where is a symmetric positive definite matrix.

Theorem 6. Let (H3.3) and (H3.4) hold; then the trivial solution of (1) is exponentially stable in the mean square. Assume that there exist symmetric positive definite matrices and a constant such that

In order to prove Theorem 6, we need three lemmas, proofs of which are left in appendix.

Lemma 7. Fix the initial data arbitrarily. Then,

for any , where is a constant larger than .

Lemma 8. Fix the initial data arbitrarily. Then,

for any .

Lemma 9. Let (H3.3) and (H3.4) hold. Then,

for any , where is a constant larger than .

Based on the above Lemmas 7–9, we now carry out a proof for Theorem 6.

Proof of Theorem 6. Fix the initial data arbitrarily. Applying Itô’s formula to , we have

Combining Lemmas 3 and (24) as well as (H3.4), we can estimate the first item of (32) as follows:

where .

By (24), the last item of (32) yields

Substituting the above two into (32), we get

For small enough , we derive

If (25) holds, then we can choose small enough such that

Applying Itô’s formula to , we derive

Substituting (35) into the above yields

Taking the expectation in the above, we have

Now we apply Lemmas 7–9 to the last three terms on the right-hand side of (40) to get an estimate of as follows:

for , where

Since is positive definite,

where is the smallest eigenvalue of .

Then,

It then follows from (41) that

Hence,

This easily yields

Then (1) is exponentially stable in the mean square.

Theorem 10. Let , under the same assumption as Theorem 6, if

Then,

Proof. let , under the same assumption as Theorem 6. It follows from (48) that

for all , . Then

For , , we have

Let be arbitrary. By Doob’s martingale inequality, it follows from (52) that

Thus, it follows from the Borel-Cantelli lemma that, for almost all , there exists , and ,

So, for , ,

This easily yields

Since is arbitrary, we must have

Remark 11. The exponentially stable in the mean square of SMDDE (1) implies the almost surely exponentially stable. In general, Theorem 10 is still true for th moment exponential stability.

Let us single out two important special cases.

Case 1. If , then (1) reduces to nonlinear deterministic differential equation with mixed delay

Applying Theorem 6 to (58), we obtain the following useful result.

Corollary 12. Let (H3.4) and condition (22) hold, then the trivial solution of (58) is exponentially stable in the mean square. Assume that there exists a symmetric positive definite matrices and a constant such that

Remark 13. The bound for the time delay when SMDDE (1) is exponentially stable follows from (25), and the bound for the corresponding deterministic case follows from (60).

Case 2. If the time delay , then (1) reduces to nonlinear SDE

Corollary 14. If SDE (61) is exponentially stable and the time delay is sufficiently small, then the corresponding SMDDE (1) will remain exponentially stable.

Proof. SMDDE (1) can be rewritten as

It is clear that if the time delay is sufficiently small, then SMDDE (1) is regarded as the perturbed system of the corresponding SDE (61) (without delay). On the other hand, the perturbation term

could be so small that the perturbed equation (1) would behave in a similar way as (61) asymptotically. Applying Theorem 5 and Remark 12 in [15], we derive SMDDE (1) which will remain exponentially stable.

4. Example

Let us now present a simple example to illustrate our results which can help us find the time delay upper limit.

Example 1. Let us now consider a two-dimensional SMDDE

where

for , in , and : represent the nonlinear uncertainties. is a given two-dimensional Brownian motion,

represent given functionals of the path segment of , is the given averaging parameter, and is a given finite time delay.

For convenience, let us choose to be the second-order identity matrix, and . We assume that, for any ,

Via a simple calculation, it is easy to estimate that

that is, , , , , , and .

By plugging these into (25), it is easy to find ; that is, if , then (64) is exponentially stable in the mean square and is also almost surely exponentially stable.

If , by plugging , , , , and into (60), then, we would conclude that (64) is exponentially stable provided .

Remark 15. Figures 1 and 2 give the simulation results of Example 1 when , (Figure 1) and , (Figure 2). The parameter values used in the calculations are ,  , and .

Figure 1: The simulation results of Example 1 when , .

Figure 2: The simulation results of Example 1 when , .

5. Concluding Remarks

In this paper, we investigate not only the exponential stability in the mean square but also the almost surely exponential stability for a class of SMDDEs based on Lyapunov stability theory, Itô formula, stochastic analysis, inequality technique, and so on. We first consider the existence and uniqueness of the adapted solution by employing fixed point theorem. Next, some sufficient conditions of exponential stability and corollaries for stochastic differential systems with mixed delays are obtained by using Lyapunov function. Theorem 10 shows that the exponentially stable in the mean square of SMDDE implies the almost surely exponentially stable. The obtained results generalize and improve some recent results (for instance, [19–21]). In particular, our theoretical results show that if SDE is exponentially stable and the time delay is sufficiently small, then the corresponding SMDDE will remain exponentially stable. Moreover, the time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice. It should be mentioned that the approach provided here is different from those used in [19–21]. Finally, we present a simple example to illustrate the effectiveness of our stable results. Another challenging problem is to study a class of stochastic differential equations with mixed variable delays or a class of stochastic control systems with correlated state and observation noises (for instance, [24]). We hope to study these problems in forthcoming papers.

Appendix

We now present proofs of Lemmas 7–9.

Proof of Lemma 7. For any , we easily get

for any , where .

Proof of Lemma 8. Similar to (26), for any , we have

For , it then follows from (17) that

Similar to (26) and substituting (26) into the above, we obtain (29); that is

where .

Substituting (29) into (A.2) yields

The relation (28) in Lemma 8 is then proved.

Proof of Lemma 9. Similar to (26), for any , we have

Substituting (27) into the above inequality yields

The relation (30) in Lemma 9 is then proved.

On the other hand, for , we have

By (22) and (23), it follows from Lemmas 3-4 that

Similar to (27), for , we have

where .

Substituting (26), (28), and (30) into (A.10), for , we get

where .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities (JBK130401).

References

S. Cong, “On exponential stability conditions of linear neutral stochastic differential systems with time-varying delay,” International Journal of Robust and Nonlinear Control, vol. 23, no. 11, pp. 1265–1276, 2013.

H. Li, Z. Lin, and B. Zhou, “An analysis of the exponential stability of linear stochastic neutral delay systems,” International Journal of Robust and Nonlinear Control, 2013.

X. Mao, “A note on the LaSalle-type theorems for stochastic differential delay equations,” Journal of Mathematical Analysis and Applications, vol. 268, no. 1, pp. 125–142, 2002.

W.-H. Chen, Z.-H. Guan, and X. Lu, “Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: an LMI approach,” Systems & Control Letters, vol. 54, no. 6, pp. 547–555, 2005.

Y. Sun and J. Cao, “th moment exponential stability of stochastic recurrent neural networks with time-varying delays,” Nonlinear Analysis: Real World Applications, vol. 8, no. 4, pp. 1171–1185, 2007.

X. Meng, M. Tian, and S. Hu, “Stability analysis of stochastic recurrent neural networks with unbounded time-varying delays,” Neurocomputing, vol. 74, no. 6, pp. 949–953, 2011.

Y. Hu, F. Wu, and C. Huang, “Robustness of exponential stability of a class of stochastic functional differential equations with infinite delay,” Automatica, vol. 45, no. 11, pp. 2577–2584, 2009.

X. Mao, “Robustness of exponential stability of stochastic differential delay equations,” IEEE Transactions on Automatic Control, vol. 41, no. 3, pp. 442–447, 1996.

X. Mao and A. Shah, “Exponential stability of stochastic differential delay equations,” Stochastics and Stochastics Reports, vol. 60, no. 1-2, pp. 135–153, 1997.

W. Zhu and J. Hu, “Stability analysis of stochastic delayed cellular neural networks by LMI approach,” Chaos, Solitons & Fractals, vol. 29, no. 1, pp. 171–174, 2006.

W. Zhu and J. Hu, “Exponential stability of stochastic recurrent neural network with time delays,” International Journal of Computational Intelligence Research, vol. 2, no. 1, pp. 52–54, 2006.

S. Xie and L. Xie, “Stabilization of a class of uncertain large-scale stochastic systems with time delays,” Automatica, vol. 36, no. 1, pp. 161–167, 2000.

W. Zhu, X. Ruan, Y. Qin, and J. Zhuang, “Exponential stability of stochastic nonlinear dynamical price system with delay,” Mathematical Problems in Engineering, vol. 2013, Article ID 168169, 9 pages, 2013.

B. Song, J. H. Park, Z.-G. Wu, and X. Li, “New results on delay-dependent stability analysis and stabilization for stochastic time-delay systems,” International Journal of Robust and Nonlinear, 2013.

Y. Hu and F. Wu, “Stochastic Kolmogorov-type population dynamics with infinite distributed delays,” Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1407–1428, 2010.

W. Wu, B. T. Cui, and X. Y. Lou, “Global exponential stability of Cohen-Grossberg neural networks with distributed delays,” Mathematical and Computer Modelling, vol. 47, no. 9-10, pp. 868–873, 2008.

L. Yin, Y. Chen, and Y. Zhao, “Global exponential stability for a class of neural networks with continuously distributed delays,” Advances in Dynamical Systems and Applications, vol. 4, no. 2, pp. 221–229, 2009.

J. Zhou, S. Li, and Z. Yang, “Global exponential stability of Hopfield neural networks with distributed delays,” Applied Mathematical Modelling, vol. 33, no. 3, pp. 1513–1520, 2009.

Q. Zhu and B. Song, “Exponential stability of impulsive nonlinear stochastic differential equations with mixed delays,” Nonlinear Analysis. Real World Applications, vol. 12, no. 5, pp. 2851–2860, 2011.

F. Deng, M. Hua, X. Liu, Y. Peng, and J. Fei, “Robust delay-dependent exponential stability for uncertain stochastic neural networks with mixed delays,” Neurocomputing, vol. 74, no. 10, pp. 1503–1509, 2011.

L. Xu and D. Xu, “Exponential -stability of impulsive stochastic neural networks with mixed delays,” Chaos, Solitons & Fractals, vol. 41, no. 1, pp. 263–272, 2009.

S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1994.

L. Chen and Z. Wu, “Maximum principle for the stochastic optimal control problem with delay and application,” Automatica, vol. 46, no. 6, pp. 1074–1080, 2010.

G. Wang, Z. Wu, and J. Xiong, “Maximum principles for forward-backward stochastic control systems with correlated state and observation noises,” SIAM Journal on Control and Optimization, vol. 51, no. 1, pp. 491–524, 2013.